A Factored Radix Number Filter

fr transforms numbers to and from quasi-decimal versions of higher base numbers.

What is a factored radix number?

A standard number system like base 20 is defined by a radix. A factored radix number system FR(P,S) is defined by two radices, for example, FR(2,10).

FR(2,10) is similar to base 20:

• It uses an alphabet of 20 digits (P*S=20), e.g., 0123456789abcdefghij.
• A number in base 20 can be represented in the same number of digits in FR(2,10).

However, FR(2,10) is also similar to decimal:

• Any number that looks like decimal retains its decimal value. For example, 426 in FR(2,10) has the decimal value 426 (whereas 426 in base 20 is decimal 1646).
• Non-decimal digits are optional and used only to reduce the number of digits.

Suppose a number is to be represented using at most 2 digits. Using decimal, 100 values are possible. Like base 20, FR(2,10) extends this range to 400 values. However, unlike base 20, the sequence runs 0,..,99, as for decimal, and then 0a,..,jj (leading zeros are significant).

Where a number in FR(2,10) has a non-decimal digit, the decimal value is found as in this example:

 d7   FR(2,10)
 37   "stem": non-decimal digits replaced with mod 10 value
 10   "prefix", base 2: a..j -> 1, 0..9 -> 0, so b7 -> 10
  2   prefix converted to base 10
237   prefix and stem concatenated: decimal equivalent of b7

For a comprehensive account, see https://mbreen.com/fr

To install and run

The filter is written in C (C99) to keep things easy. On a typical installation of Linux this should work:

make                        # create ./fr
./fr                        # get a usage message

To see the full sequence of 2-digit FR(2,10) numbers mentioned above and their decimal equivalents (using bash):

paste <(seq 0 399 |FRDIGITS=DCA fr 2) <(seq 0 399) |less